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Let $D \subseteq \mathbb{P}^{2}$ be the union of three distinct lines which intersect at a point $x,$ and let $C \subseteq \mathbb{P}^{2}$ be a smooth plane cubic which intersects $D$ in 9 points, none of which is $x.$ The blowup $X$ of $\mathbb{P}^{2}$ at these 9 points is smooth, and admits a morphism to $\mathbb{P}^{1}$ whose fibers are the members of the pencil of cubics spanned by $C$ and $D.$ One of these fibers is (the strict transform of) $D,$ which has an ordinary triple point.

EDIT: To get an example containing a rational curve with an ordinary triple point singularity, we can apply this construction with $D$ a plane quartic having an ordinary triple point $x$ and $C$ a smooth plane quartic intersecting $D$ at 16 points, none of which is $x.$ The blowup $X$ of $\mathbb{P}^{2}$ at $C \cap D$ is then a surface of the desired type.